University of Montpellier (France)

A school organized under the auspices of the French Mathematical Society «Etats de la recherche» program and within the French national centre for Scientific Research (CNRS) «Thematic schools» framework.

The goal of the school is to introduce a broad audience of mathematicians and young researchers to a selection of currently very active fields of research. It will be centered on themes of high current interest in mathematical relativity, with a strong emphasis on the differential geometric side of the theory. Ph. D students are especially invited to attend.

Building #10

Science Campus, University of Montpellier

Introduction to the mathematical theory of black holes (lecture 1)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

In the lectures I plan to discuss the geometry of spherically symmetric black holes, that of the Kerr black hole,
and of the Emperan-Reall black rings. Conformal and projection diagrams will be discussed, and some elements of the theory of uniqueness of stationary black holes will be presented.
The lectures will be based on selected chapters of the monograph "Geometry of black holes", available at http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black%20Holes/BlackHolesViennaJanuary2015.pdf

Orateur:
Piotr Chrusciel
(University of Vienna)

14:20
→
15:20

On the notion of quasi local mass in General Relativity (lecture 1)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

Contents:
(1) Review of mass and energy in relativity.
(2) ADM mass and energy-momentum.
(3) Various definitions of quasi-local mass.
A set of notes is provided for the 4 lectures in the Abstracts section

Orateur:
Mu-Tao Wang
(Columbia University, NY)

Paper

15:20
→
15:50

Coffee break
30m
Building #10 (Science Campus, University of Montpellier)

Building #10

Science Campus, University of Montpellier

15:50
→
16:50

Horizons in General Relativity (lecture 1)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

In the first two lectures I will describe the basic results on the significance, existence, and properties of apparent horizons (or more precisely “marginally outer trapped surfaces”) in initial data sets. Taking into account the preferences of the audience, I will then sketch the proofs of one or two fundamental results in mathematical relativity that build on this theory. The possibilities include the minimal surface proof of the Riemannian positive energy theorem, the marginally outer trapped surface proof of the spacetime positive mass theorem, and the existence of black holes due to condensation of matter.

Introduction to the mathematical theory of black holes (lecture 2)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

In the lectures I plan to discuss the geometry of spherically symmetric black holes, that of the Kerr black hole,
and of the Emperan-Reall black rings. Conformal and projection diagrams will be discussed, and some elements of the theory of uniqueness of stationary black holes will be presented.
The lectures will be based on selected chapters of the monograph "Geometry of black holes", available at http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black%20Holes/BlackHolesViennaJanuary2015.pdf

Building #10

Science Campus, University of Montpellier

On the center of mass in General Relativity (short talk)40mLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

In many situations in Newtonian Gravity, understanding the motion of the
center of mass of a system is key to understanding the general "trend"
of the motion of the system. It is thus desirable to also devise a
notion of center of mass with similar properties in General Relativity.
However, while the definition of the center of mass via the mass density
is straightforward in Newtonian Gravity, there is a priori no definitive
corresponding notion in General Relativity. Instead, there are several
alternative approaches to defining the center of mass of a system. We
will discuss some of these different approaches for both asymptotically
Euclidean and asymptotically hyperbolic systems and present some new
ideas as well as explicit (counter-)examples.

Orateur:
Carla Cederbaum
(Tübingen University)

12:00
→
13:00

On the notion of quasi local mass in General Relativity (lecture 2)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

Contents:
(1) New definition of quasi-local energy.
(2) Isometric embedding into the Minkowski space.
(3) The proof of positivity.
A set of notes is provided for the 4 lectures in the Abstracts section

Orateur:
Mu-Tao Wang
(Columbia University, NY)

Paper

15:00
→
16:00

Horizons in General Relativity (lecture 2)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

In the first two lectures I will describe the basic results on the significance, existence, and properties of apparent horizons (or more precisely “marginally outer trapped surfaces”) in initial data sets. Taking into account the preferences of the audience, I will then sketch the proofs of one or two fundamental results in mathematical relativity that build on this theory. The possibilities include the minimal surface proof of the Riemannian positive energy theorem, the marginally outer trapped surface proof of the spacetime positive mass theorem, and the existence of black holes due to condensation of matter.

Building #10

Science Campus, University of Montpellier

On the notion of quasi local mass in General Relativity (lecture 3)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

Contents:
(1) Variation of quasi-local energy.
(2) The optimal embedding equation.
(3) Solving the optimal isometric embedding equation at spatial and null infinity.
A set of notes is provided for the 4 lectures in the Abstracts section.

Introduction to the mathematical theory of black holes (lecture 3)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

In the lectures I plan to discuss the geometry of spherically symmetric black holes, that of the Kerr black hole,
and of the Emperan-Reall black rings. Conformal and projection diagrams will be discussed, and some elements of the theory of uniqueness of stationary black holes will be presented.
The lectures will be based on selected chapters of the monograph "Geometry of black holes", available at http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black%20Holes/BlackHolesViennaJanuary2015.pdf

Building #10

Science Campus, University of Montpellier

On the notion of quasi local mass in General Relativity (lecture 4)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

Contents:
(1) Minimizing and rigidity property of critical points of quasilocal energy.
(2) Quasilocal angular momentum and center of mass and their limits at infinity.
(3) Asymptotically hyperbolic initial data sets.
A set of notes is provided for the 4 lectures in the Abstracts section

Lecture room SC 10.01, building #10, Science Campus

University of Montpellier

The stability of the Einstein-Lichnerowicz equation is defined as the continuous dependence of the set of its positive solutions in the choice of the background physics data of the conformal method. When the conditions ensuring stability fail, surprising phenomena can arise, such as the existence of an infinite number of concentrating positive solutions. In this talk we will investigate some of these instability phenomena for the Einstein-Lichnerowicz equation when a non-trivial scalar field is present.

Orateur:
Bruno PREMOSELLI
(Université de Cergy-Pontoise)

15:00
→
16:00

Horizons in General Relativity (lecture 3)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

In the first two lectures I will describe the basic results on the significance, existence, and properties of apparent horizons (or more precisely “marginally outer trapped surfaces”) in initial data sets. Taking into account the preferences of the audience, I will then sketch the proofs of one or two fundamental results in mathematical relativity that build on this theory. The possibilities include the minimal surface proof of the Riemannian positive energy theorem, the marginally outer trapped surface proof of the spacetime positive mass theorem, and the existence of black holes due to condensation of matter.

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

Localized solutions of the Einstein equations: a few words about their geometry and physics (short talk)40mLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

It is a truly surprising fact that the Einstein constraint equations own an overabundance of localized solutions, namely solutions that coincide with arbitrarily-assigned data inside a given solid cone and are trivial outisde of a cone of slightly larger angle. In this talk, I will briefly present them and comment on their physical relevance (related to gravitational shielding phenomena) as well as on their geometric content (which concerns the link with the isoperimetric problem, large outlying CMC spheres and stable minimal surfaces). This is mostly based on joint work with Richard Schoen.

Orateur:
Alessandro CARLOTTO
(ETH Zürich)

17:15
→
18:00

Open problem session and discussionLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

Horizons in General Relativity (lecture 4)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

In the first two lectures I will describe the basic results on the significance, existence, and properties of apparent horizons (or more precisely “marginally outer trapped surfaces”) in initial data sets. Taking into account the preferences of the audience, I will then sketch the proofs of one or two fundamental results in mathematical relativity that build on this theory. The possibilities include the minimal surface proof of the Riemannian positive energy theorem, the marginally outer trapped surface proof of the spacetime positive mass theorem, and the existence of black holes due to condensation of matter.

Building #10

Science Campus, University of Montpellier

Introduction to the mathematical theory of black holes (lecture 4)1hLecture room SC 10.01, building #10 (Science Campus, University of Montpellier)

Lecture room SC 10.01, building #10

Science Campus, University of Montpellier

In the lectures I plan to discuss the geometry of spherically symmetric black holes, that of the Kerr black hole,
and of the Emperan-Reall black rings. Conformal and projection diagrams will be discussed, and some elements of the theory of uniqueness of stationary black holes will be presented.
The lectures will be based on selected chapters of the monograph "Geometry of black holes", available at http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black%20Holes/BlackHolesViennaJanuary2015.pdf

Orateur:
Piotr Chrusciel
(University of Vienna)

12:15
→
12:30

Closing of the School
15m
Building #10 (Science Campus, University of Montpellier)